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Thermodynamics, ideal gas, virial coefficient

  1. Jul 12, 2013 #1
    1. The problem statement, all variables and given/known data
    So the problem gives you the compressibility factor Z. It gives two equations, one for ideal gas and other one for non-ideal gas.
    The question states:
    The second Virial coefficient B(T) depends on the temperature T and is related to the intermolecular potential B(T) = 2[itex]\pi\N∫[1-exp(-ψ(r)/kT)]r^2 dr

    boundary is 0 ~ ∞.

    Derive the second Virial coefficient B(T) when the dependence of the intermolecular potential ψ(r) on molecular distance r is given as follows,

    r < σ : ψ(r) = ∞,
    r >= σ : ψ(r) = -ε(σ/r)^6

    2. Relevant equations
    Z = 1 + (n/V)*B(T) + (n/V)^2 * C(T) + (n/V)^3 * D(T)+...
    for ideal gas, it is : Z = PV / nRT.
    B(T), C(T), and D(T) are the second, the third, and the fourth virial coefficients respectively. P is a pressure, V is a volume, n is the number of moles, R is the gas constant, and T is a temperature. It also gives me a equation with Boltzmann constant k and Avogadro's number N: R = kN.


    3. The attempt at a solution
    Since the problem have given us different ψ(r) when the boundary of r is different, I decided to separate the integral into two parts also, one with boundary of 0~σ and the second one with σ~∞. From there I decided to substitute the ψ(r) into corresponding integrals, one ∞ into the one with boundaries of 0~σ and -ε(σ/r)^6 into the one with σ~∞. I thought I can derive from this but I seem to get stuck when I'm deriving the integral with the boundary of σ~∞. Please give me some help me with some advice!
     
  2. jcsd
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